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Molecular gas in absorption and emission along the line of sight to W31C G10.62-0.38

We used the ARO 12m antenna to observe emission from the J=1-0 lines of carbon monoxide, \hcop\ and HNC and the J=2-1 line of CS toward and around the continuum peak used for absorption studies and we compare them with CH, HNC, C\p\ and other absorption spectra from PRISMAS. We develop a kinematic analysis that allows a continuous description of the spectral properties and relates them to viewing geometry in the Galaxy. As for CH, HF, C\p, \hcop\ and other species observed in absorption, mm-wave emission in CO, \hcop, HNC and CS is continuous over the full velocity range expected for material between the Sun and W31 4.95 kpc away. CO emission is much stronger than average in the Galactic molecular ring and the mean \HH\ density derived from CH, $4 \pccc \la$ 2$<$n(\HH)$>$ $\la 10 \pccc$ at 4 $\la$ R $\la$ 6.4 kpc, is similarly elevated. The CO-\HH\ conversion factor falls in a narrow range \XCO\ $= 1-2\times10^{20}~\HH\ \pcc~({\rm K}-\kms)^{-1}$ if the emitting gas is mostly on the near side of the sub-central point, as we suggest. The brightnesses of \hcop, HNC, and CS are comparable (0.83\%, 0.51\% and 1.1\% respectively relative to CO) and have no variation in galactocentric radius with respect to CO. Comparison of the profile-averaged \hcop\ emission brightness and optical depth implies local densities n(H) $\approx 135\pm25\pccc$ with most of excitation of \hcop\ from electrons. At such density, a consistent picture of the \HH-bearing gas, accounting also for the CO emission, has a volume filling factor 3\% and a 5 pc clump or cloud size.